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Questions and Answers

A vector is linearly dependent if it is equal to the zero vector.

True

Two vectors are linearly dependent if one is a multiple of the other.

True

If you perform Gaussian elimination and find a line with all zeros, it indicates linear dependence.

True

What is the relationship between linear mappings and linear independence?

Linear mappings preserve linear independence if the map is injective. Signup and view all the answers

Can a subset of independent vectors generate a vector space?

No, it cannot. Signup and view all the answers

A vector space V is finitely generated if there exists an n and a surjective mapping T: F^n → W.

True Signup and view all the answers

What defines a base of a vector space?

A base consists of independent and generating vectors. Signup and view all the answers

What does 'maximal-independent' mean for a base?

No additional vector can be added without losing independence. Signup and view all the answers

Bijective linear mappings bring bases into bases.

True Signup and view all the answers

In a space V of dimension n, which of the following statements is true?

All of the above. Signup and view all the answers

What does it mean for two vector spaces V and W to be isomorphic?

V is isomorphic to W if dimV = dimW. Signup and view all the answers

More than one linear mapping can send a basis of an n-dimensional vector space V to any n vectors in another vector space W.

False Signup and view all the answers

What does the dual space V' consist of?

It is the set of all linear forms on V. Signup and view all the answers

What is the basis of the dual space?

The dual basis consists of functions that map coordinates. Signup and view all the answers

What is the transposition of a linear mapping?

It is a function that creates a linear form on V from a linear form on W by composition. Signup and view all the answers

What is the annihilator of a subspace?

It is the set of all linear forms where f(x) = 0 for all x in M. Signup and view all the answers

The _ of the transpose is the annihilator of the _

kernel, image Signup and view all the answers

Complete the theorem: If T: V → W is a linear mapping, then T is surjective ⇔ T' is ______.

injective Signup and view all the answers

What is the bidual of V?

V'' Signup and view all the answers

Why is the canonical embedding important?

It builds an isomorphism between V and V''. Signup and view all the answers

What's an operation?

0 holds: a#b is in A Signup and view all the answers

What's a semigroup?

Property 0: a#b is in a; Property 1: (a#b)#c = a#(b#c) Signup and view all the answers

What's a monoid?

Property 0: a#b is in a; Property 1: (a#b)#c = a#(b#c); Property 2: It exists a unit such that: a#u=a Signup and view all the answers

What's a group?

Property 0: a#b is in a; Property 1: (a#b)#c = a#(b#c); Property 2: It exists a unit such that: a#u=a; Property 3: It exists an inverse such that: a#ã=u Signup and view all the answers

What's an abelian group?

Property 0: a#b is in A; Property 1: (a#b)#c = a#(b#c); Property 2: It exists a unit such that: a#u=a; Property 3: It exists an inverse such that: a#ã=u; Property 4: a#b = b#a Signup and view all the answers

What's a field? Give some examples.

F is a field if (F, +) is an abelian group and (F*, *) is an abelian group. Examples of fields are: B, Q, R, C. Signup and view all the answers

What does it mean that b is divisible by a?

Means that we can write it as b = aq for some q in the integral domain D. We write a | b. Signup and view all the answers

Prove that the remainder and the quotient are unique.

If a = bq + r = bq' + r', then r - r' = b(q' - q) implies r = r' and q = q'. Signup and view all the answers

What does it mean that a ~ b (mod m)? Prove it.

It means that m | (a - b). It means that (a - b) = mk for some k in Z. Signup and view all the answers

What's an equivalence relation?

A relation that is reflexive, symmetric, and transitive. Signup and view all the answers

What is a relation? Give examples.

A relation is a subset of pairs of elements from a set. For example, a >= b is a relation. Signup and view all the answers

Prove that any non-empty set of integers closed under addition and subtraction either consists of zero alone or else contains a least positive element.

If the set is not empty, it exists at least one element a. The process finds all multiples of a. Signup and view all the answers

What is a vector?

A vector can be considered as a function that associates a certain number of slots with a certain number of values of the field. Signup and view all the answers

How are operations between vectors defined?

They are defined point-wise, such as scalar multiplication and sum. Signup and view all the answers

When is a subset of F^S said to be linear?

Whenever all possible linear combinations in that subset are still in the subset. Signup and view all the answers

What is a vector space?

V is a vector space over a field F if it satisfies properties like closure under addition and scalar multiplication. Signup and view all the answers

Can you mention some examples of vector spaces?

Some examples are: continuous functions, polynomials, vectors in R^2 and R^3. Signup and view all the answers

Prove that the zero of a vector space is unique.

If k + x = x, then by canceling x we find k = 0 for every x. Signup and view all the answers

Define linear combination.

A vector x is a linear combination of vectors (x1, ..., xn) if x = c1x1 + ... + cnxn. Signup and view all the answers

What is a linear subspace of a vector space?

A vector linear subspace M satisfies properties like closure under addition and scalar multiplication. Signup and view all the answers

Is a subspace always a vector space itself? What is the smallest vector linear subspace you know?

Yes, it is. The smallest vector space is the trivial vector space which consists of the zero vector. Signup and view all the answers

Given two subspaces M and N, define M+N and M ∩ N.

M+N = {x+y: x in M and y in N}; M ∩ N = {x in V: x in M and x in N}. Signup and view all the answers

What is a linear mapping?

A mapping T:V→W is linear if T(x+y) = Tx + Ty and T(cx) = cT(x). Signup and view all the answers

What is the definition of linear form?

A linear form is a linear mapping f:V→F, where F is regarded as a vector space over itself. Signup and view all the answers

Prove that given two subspaces M and N and a linear mapping T:V→W, T(M) and T(N) are both subspaces.

T(0) = 0. For any y, y' in T(M), T(y+y') = Ty + Ty'. Signup and view all the answers

Define the kernel and the image of a linear mapping T.

The kernel is KerT = {x in V : Tx = 0}; the image is ImT = {Tx : x in V}. Signup and view all the answers

What properties do you get from the KerT?

For x, x' in V, Tx = Tx' IFF x - x' in KerT; KerT = {0} IFF T is injective. Signup and view all the answers

If V and W are vector spaces over F, define the operations of L(V, W) and prove it.

Define sum as (S+T)x = Sx + Tx and scalar multiplication as (aT)x = a(Tx). Signup and view all the answers

What does it mean that V is isomorphic to W?

It means there exists a bijection T: V→W. Signup and view all the answers

Prove that being isomorphic is an equivalence relation.

Reflexivity: identity mapping; Symmetry: inverse mapping; Transitivity: composition of bijections. Signup and view all the answers

What is an equivalence relation in a set X?

A relation that satisfies reflexivity, symmetry, and transitivity. Signup and view all the answers

Give at least three fundamental examples of equivalence classes.

<ol> <li>f:A→B having the same image; 2. x - y in M for vectors; 3. Cosets of vector subspaces.</li> </ol> Signup and view all the answers

What is a partition? What are its properties?

A partition is a set of non-empty subsets where each element belongs to exactly one subset. Signup and view all the answers

Define quotient set and quotient mapping.

The quotient set X/~ contains all equivalence classes; the quotient mapping assigns every x to its equivalence class. Signup and view all the answers

What is the quotient vector space?

Given V and a subspace M, V/M is defined using sum and scalar multiplication from V. Signup and view all the answers

What does the first isomorphism theorem say?

N is a linear subspace of V; T(V) is a linear subspace of W; V/KerT is isomorphic to Im(V). Signup and view all the answers

How is linear span defined? What are its characteristics?

The linear span of a set A is the set of all possible linear combinations of its elements. Signup and view all the answers

Prove that if T:V→W is linear and A is a subset of V, then T([A]) = [T(A)].

T([A]) = T(c1x1 + ... + cnxn) = c1T(x1) + ... + cnT(xn) = [T(A)]. Signup and view all the answers

Prove that surjective linear mappings bring generators into generators.

If the map is surjective, T(A) generates W, implying T(A) is generating. Signup and view all the answers

What does it mean that a finite list of vectors is linearly dependent?

A list of n vectors is dependent if coefficients exist such that c1x1 + ... + cnxn = 0, with at least one non-zero. Signup and view all the answers

What can you say about a map T:F^n→W that is defined by T(a1, a2, ..., an) → a1x1 + ... + anxn when the chosen {x1, x2, ..., xn} are linearly dependent?

T is not injective, as there exists coefficients making T equal to zero. Signup and view all the answers

When can you quickly recognize that a list of vectors is linearly dependent?

If there exists a linear combination of them that gives you the zero vector with non-trivial coefficients. Signup and view all the answers

Study Notes

Operations in Algebra

An operation is defined as a function where for any elements a and b from a set A, the result a # b is also in A.

Algebraic Structures

A semigroup requires closure under an operation and associativity: (a # b) # c = a # (b # c).
A monoid adds the existence of an identity element, u, such that a # u = a.
A group extends a monoid with the necessity of inverses: for every element a, there is an element ã such that a # ã = u.
An abelian group is a group where the operation is commutative: a # b = b # a.

Fields and Divisibility

A field requires two operations (+ and ·), with both structures being abelian groups and the existence of multiplicative inverses.
Examples of fields include B (binary), Q (rational numbers), R (real numbers), and C (complex numbers).
Divisibility means b can be expressed as b = aq for some q in the integers, denoted a | b.

Modular Arithmetic

Congruence a ~ b (mod m) means m divides (a - b).
It implies that a and b share the same remainder when divided by m.

Relations and Equivalence

An equivalence relation is reflexive, symmetric, and transitive.
A relation on a set L consists of ordered pairs from L x L.
The equivalence class [x] includes all elements that are equivalent to x under a given relation.

Vector Spaces

A vector can be viewed as a function mapping indices to field values, defined in F^S.
Vector operations are pointwise: scalar multiplication and vector addition.
A linear subspace respects the closed nature of vector operations and contains the zero vector.

Properties of Vector Spaces

A vector space must satisfy properties including closure under addition and scalar multiplication, associativity, commutativity, existence of additive identity and inverses.
A linear combination involves scalars combining vectors to form another vector in the same space.

Linear Mappings

A linear mapping T: V → W requires T(x + y) = T(x) + T(y) and T(cx) = cT(x).
Kernel is the set of vectors that map to zero, while image refers to the range of mappings.

Span and Independence

The span of a set A includes all linear combinations, forming the smallest subspace containing A.
Vectors are linearly dependent if a non-trivial combination of them equals zero; otherwise, they are independent.
A collection of vectors can generate a vector space if they span it.

Quotient Spaces and Isomorphisms

A quotient vector space V/M is formed by defining operations using representatives from equivalence classes.
Isomorphism between V and W indicates a bijective mapping preserving structure; two spaces are isomorphic if such a mapping exists.

Fundamental Definitions

Partition divides a set into non-empty, disjoint subsets, with every element belonging to one subset.
The first isomorphism theorem relates the kernel and image of a linear mapping, linking the quotient of vector spaces.

Bases and Dimension

A basis of a vector space is a set of vectors that are independent and generate the space, and a canonical basis uses standard unit vectors.
The mapping of dimensions links finite generation to the existence of surjective mappings onto vector spaces.

Maximal Independence and Minimal Generating Sets

A basis is maximal independent when adding any new vector to the basis makes it dependent.
It is minimally generating if removing any vector from the basis causes the set to no longer generate the space.### Linear Independence and Bases
Adding another vector ( k ) to a list of vectors ( {x_1, x_2, \ldots, x_n} \ causes it to lose linear independence.
Removing the last vector ( x_n ) from a list results in a set ( {x_1, x_2, \ldots, x_{n-1}} ) that does not generate the entire space.

Bijective Linear Mappings

Bijective linear mappings ( T: V \to W ) transform bases in vector space ( V ) into bases in vector space ( W ).
Injective (one-to-one) mappings preserve linear independence among vectors.
Surjective (onto) mappings ensure that generating sets stay generating.

Properties of Independent and Generating Lists

In a vector space ( V ) of dimension ( n ):
- An independent list of vectors can have a maximum length of ( n ).
- A generating list must have at least ( n ) vectors.
- Lists that are independent or generating, and are exactly of length ( n ), are bases.

Isomorphism and Dimension

Two vector spaces ( V ) and ( W ) are isomorphic if and only if ( \text{dim } V = \text{dim } W ).
If the spaces are isomorphic, they have a linear bijection, ensuring equal dimension.

Uniqueness of Linear Mappings

A linear mapping that sends a basis from an ( n )-dimensional space ( V ) to any ( n ) vectors in ( W ) is unique.
It is possible to extend a mapping by completing a basis, mapping additional elements to zero.

Dual Spaces

The dual space ( V' ) consists of all linear forms ( L(V, F) ) and has a dimension equal to that of ( V ).
No natural isomorphism exists between the vector space and its dual space.

Dual Basis

The dual basis for ( (F^n)' ) is composed of functions ( f_i ) such that ( f_i(a_1, \ldots, a_n) = a_i ), where each function corresponds to an individual coordinate.

Transpose of Linear Mappings

For a linear mapping ( T: V \to W ), the transpose ( T' ) is defined by composing with linear forms on ( W ).
The transpose retains linearity and has the same rank as the original mapping.

Annihilator of a Subspace

The annihilator ( M^\circ ) of a subspace ( M \subset V ) consists of linear forms that evaluate to zero on ( M ).
The dimension of the annihilator is given by ( \text{dim } M^\circ = \text{dim } V - \text{dim } M ).

Properties of the Transpose

The kernel of the transpose ( T' ) is the annihilator of the image ( T(V) ).

Theorems Relating to Linear Mappings

If ( T: V \to W ) is a linear mapping:
- ( T ) is surjective if and only if ( T' ) is injective.
- ( T ) is injective if and only if ( T' ) is surjective.

Bidual and Canonical Embedding

The bidual of a space ( V ) is defined as ( (V')' = V'' ).
The canonical embedding relates each vector in ( V ) to its evaluation mapping in the dual space.

Importance of the Canonical Embedding

The canonical embedding establishes an isomorphism between ( V ) and ( V'' ).
This mapping is linear, surjective, and injective, confirming that ( \text{Ker } S = {\theta} ).

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Questions and Answers

A vector is linearly dependent if it is equal to the zero vector.

Two vectors are linearly dependent if one is a multiple of the other.

If you perform Gaussian elimination and find a line with all zeros, it indicates linear dependence.

What is the relationship between linear mappings and linear independence?

Can a subset of independent vectors generate a vector space?

A vector space V is finitely generated if there exists an n and a surjective mapping T: F^n → W.

What defines a base of a vector space?

What does 'maximal-independent' mean for a base?

Bijective linear mappings bring bases into bases.

In a space V of dimension n, which of the following statements is true?

What does it mean for two vector spaces V and W to be isomorphic?

More than one linear mapping can send a basis of an n-dimensional vector space V to any n vectors in another vector space W.

What does the dual space V' consist of?

What is the basis of the dual space?

What is the transposition of a linear mapping?

What is the annihilator of a subspace?

The _____ of the transpose is the annihilator of the _____

Complete the theorem: If T: V → W is a linear mapping, then T is surjective ⇔ T' is ______.

What is the bidual of V?

Why is the canonical embedding important?

What's an operation?

What's a semigroup?

What's a monoid?

What's a group?

What's an abelian group?

What's a field? Give some examples.

What does it mean that b is divisible by a?

Prove that the remainder and the quotient are unique.

What does it mean that a ~ b (mod m)? Prove it.

What's an equivalence relation?

What is a relation? Give examples.

Prove that any non-empty set of integers closed under addition and subtraction either consists of zero alone or else contains a least positive element.

What is a vector?

How are operations between vectors defined?

When is a subset of F^S said to be linear?

What is a vector space?

Can you mention some examples of vector spaces?

Prove that the zero of a vector space is unique.

Define linear combination.

What is a linear subspace of a vector space?

Is a subspace always a vector space itself? What is the smallest vector linear subspace you know?

Given two subspaces M and N, define M+N and M ∩ N.

What is a linear mapping?

What is the definition of linear form?

Prove that given two subspaces M and N and a linear mapping T:V→W, T(M) and T(N) are both subspaces.

Define the kernel and the image of a linear mapping T.

What properties do you get from the KerT?

If V and W are vector spaces over F, define the operations of L(V, W) and prove it.

What does it mean that V is isomorphic to W?

Prove that being isomorphic is an equivalence relation.

What is an equivalence relation in a set X?

Give at least three fundamental examples of equivalence classes.

What is a partition? What are its properties?

Define quotient set and quotient mapping.

What is the quotient vector space?

What does the first isomorphism theorem say?

How is linear span defined? What are its characteristics?

Prove that if T:V→W is linear and A is a subset of V, then T([A]) = [T(A)].

Prove that surjective linear mappings bring generators into generators.

What does it mean that a finite list of vectors is linearly dependent?

What can you say about a map T:F^n→W that is defined by T(a1, a2, ..., an) → a1x1 + ... + anxn when the chosen {x1, x2, ..., xn} are linearly dependent?

When can you quickly recognize that a list of vectors is linearly dependent?

Study Notes

Operations in Algebra

Algebraic Structures

Fields and Divisibility

Modular Arithmetic

Relations and Equivalence

Vector Spaces

Properties of Vector Spaces

Linear Mappings

Span and Independence

Quotient Spaces and Isomorphisms

Fundamental Definitions

Bases and Dimension

Maximal Independence and Minimal Generating Sets

The _ of the transpose is the annihilator of the _